Proof. (i) The equality (1) takes forms $h^{1,0}=h^{0,1}$ and $h^{1,1}+2h^{2,0}=2h^{0,2}+h^{1,1}$ for $n=1,2$ respectively.

(ii) The sum of the left-hand side and the right-hand side of (1) is, of course, even and is also equal to $n\cdot (h^{0,n}+h^{1,n-1}+\cdots +h^{n,0})=n\cdot \dim _{K}H^{n}_{\mathrm {dR}}(X/K)$, so $\dim _K H^{n}_{\mathrm {dR}}(X/K)$ is even for odd $n$.

Proof. For the convenience of the reader we recall the proof in the case when $Y$ is projective over an arbitrary perfect field or arbitrary proper defined over a finite field.

If $Y$ is projective, denote by $L\in H^{2}_{\mathrm {cris}}(Y/W(k))$ the crystalline Chern class of an ample line bundle on $Y$. The hard Lefschetz theorem for crystalline cohomology ([Reference Katz and MessingKM74, Corollary 1(2)] in the case of a finite base field) shows that cup-product with $L^{d-\min (n,2d-n)}$ induces an isomorphism $H^{n}_{\mathrm {cris}}(Y/W(k))[{1}/{p}](n-d)\simeq H^{2d-n}_{\mathrm {cris}}(Y/W(k))[{1}/{p}]$ of $\varphi$-modules over $K_0$. On the other hand, Poincaré duality provides an isomorphism $H^{2d-n}_{\mathrm {cris}}(Y/W(k))[{1}/{p}]\simeq H^{n}_{\mathrm {cris}}(Y/W(k))^{\vee }[{1}/{p}](-d)$. Combining these isomorphisms we get $H^{n}_{\rm cris}(Y/W(k))[{1}/{p}](n)\simeq H^{n}_{\mathrm {cris}}(Y/W(k))[1/p]^{\vee }$. Since $t_N(D^{\vee }(-n))=-t_N(D(n))=-t_N(D)+n\cdot \dim _{K_0}D$ we get the desired equality.

If $Y$ is not projective but is defined over a finite field $k={\mathbb {F}}_{q}$ the result follows from Weil conjectures: the multiset of $q$-power Frobenius eigenvalues consists of algebraic integers and is stable under complex conjugation. Hence, it is stable under the operation $\alpha \mapsto q^{n}\alpha ^{-1}=\bar {\alpha }$ so the multiset of slopes is stable under $\lambda \mapsto n-\lambda$, as desired.

Proof of Proposition 2.1 Consider the filtered $\varphi$-module given by $D=H^{n}_{\mathrm {cris}}({\mathfrak X}_k/W(k))[{1}/{p}]$ with the Frobenius structure induced by $\mathop {Fr}:{\mathfrak X}_k \to {\mathfrak X}_k\times _{k,\sigma }k$ and the filtration on $D_K$ is given by the Hodge filtration on the de Rham cohomology of the generic fiber $H^{n}_{\mathrm {dR}}(X/K)\simeq H^{n}_{\mathrm {cris}}({\mathfrak X}/W(k))\otimes _{W(k)}K=D_K$.

By the crystalline comparison theorem proved in [Reference Bhatt, Morrow and ScholzeBMS18, Theorem 1.1(i)] for smooth proper formal schemes, the filtered $\varphi$-module $D$ is obtained by applying the functor $D_{\mathrm {cris}}$ to the $p$-adic Galois representation $H^{n}_{\mathrm {et}}(X_{\bar {K}},{\mathbb {Q}}_p)$. In particular, $D$ is weakly admissible by [Reference FontaineFon79, Proposition 4.4.5].

Hence, $t_H(D)=t_N(D)$ so by Lemma 2.3 we get $t_H(D)=n\cdot \dim _K H^{n}_{\mathrm {dR}}(X/K)-t_H(D)$. As $t_H(D)=h^{1,{n-1}}+2h^{2,n-2}+\cdots + n\cdot h^{n,0}$ and $\dim _K H^{n}_{\mathrm {dR}}(X/K)=h^{0,n}+h^{1,n-1}+\cdots + h^{n,0}$ we get the desired equality.

Proof. Let $\zeta _l$ be a primitive $l$th root of unity in $\bar {\mathbb {Q}}$. By Honda–Tate theory, there exists an abelian variety $A'$ (unique up to isogeny) over $\kappa$ with eigenvalues of the $p^{2}$-Frobenius endomorphism on first étale cohomology given by the conjugates of the Weil number $p\cdot \zeta _l$. Since $[{\mathbb {Q}}_p(\mu _l):{\mathbb {Q}}_p]=\operatorname {{ord}}_{({\mathbb {Z}}/l)^{\times }}p$ is even we have $2\dim A'=[{\mathbb {Q}}(\mu _l):{\mathbb {Q}}]=l-1$ by [Reference TateTat71, Theoreme 1(ii)]. If $\varphi _{A'}$ denotes the $p^{2}$-Frobenius endomorphism of $A'$ then $\varphi _{A'}^{l}$ acts by $p^{l}$ on $H^{1}_{\mathrm {et}}(A'_{\bar {{\mathbb {F}}}_p},{\mathbb {Z}}_t)$ (where $t$ is any prime different from $p$) so $\varphi _{A'}^{l}$ is equal to the multiplication-by-$p^{l}$ endomorphism of $A'$. Hence, $\varphi _{A'}$ induces the action of the subalgebra ${\mathbb {Z}}[p\zeta _l]\subset {\mathbb {Z}}[\mu _l]$ on the abelian variety $A'$ with $p\zeta _l$ acting by $\varphi _{A'}$.

[Reference Chai, Conrad and OortCCO14, Proposition 1.7.4.4] shows that the Serre's tensor product $A:={\mathbb {Z}}[\zeta _l]\otimes _{{\mathbb {Z}}[p\zeta _l]}A'$ is an abelian variety isogenous to $A'$ that has an action of $G$ with $\sigma$ acting as $\zeta _l$. The eigenvalues of $\sigma$ on $H^{1}_{\mathrm {cris}}(A/W(\kappa ))$ are equal to the eigenvalues of $\varphi _A$ divided by $p$ and, since characteristic polynomials of $\varphi _A$ on crystalline and étale cohomology are equal, the eigenvalues of $\sigma$ are the $l-1$ Galois conjugates of $\zeta _l$, as desired.

Proof. (i) Denote by $A^{(1)}=A\times _{\kappa ,\mathop {Fr}}\kappa$ the Frobenius-twist of $A$. For a vector space $W$ over $\kappa$ denote as well by $W^{(1)}:=W\otimes _{\kappa ,\mathop {Fr}}\kappa$ its twist. Note that the double twist $A^{(2)}:=(A^{(1)})^{(1)}$ is canonically identified with $A$.

The de Rham cohomology $H^{1}_{\mathrm {dR}}(A/\kappa )$ carries the Hodge filtration $H^{1}_{\mathrm {dR}}(A/\kappa )=F^{0}\supset F^{1}\supset F^{2}=0$ and the conjugate filtration $H^{1}_{\mathrm {dR}}(A/\kappa )=G_2\supset G_1\supset G_0=0$ satisfying $F^{0}/F^{1}=H^{0}(A,\Omega ^{1}_{A/\kappa })$, $F^{1}/F^{2}=H^{1}(A,{\mathcal {O}})$, $G_2/G_1=H^{0}(A^{(1)}$, $\Omega ^{1}_{A^{(1)}/\kappa }), G_1/G_0=H^{1}(A^{(1)},{\mathcal {O}})$, see [Reference KatzKat70, § 7]. The compositions

\[ H^{1}(A^{(1)},{\mathcal{O}})=G_1\to H^{1}_{\mathrm{dR}}(A/\kappa)\to F^{0}/F^{1}=H^{1}(A,{\mathcal{O}}) \]

and

\[ H^{1}_{\mathrm{dR}}(A^{(1)}/\kappa)=H^{1}_{\mathrm{dR}}(A/\kappa)^{(1)}\to (F^{0}/F^{1})^{(1)} = H^{1}(A,{\mathcal{O}})^{(1)}=G_1/G_0\to H^{1}_{\mathrm{dR}}(A/\kappa) \]

are both induced by the relative Frobenius morphism $\mathop {Fr}_A:A\to A^{(1)}$.

Since $\mathop {Fr}_A^{2}$ is equal to the multiplication by $p$ endomorphism $[p]_A$ up to composition with an automorphism, $\mathop {Fr}_A^{2}$ induces the zero map on $H^{1}_{\mathrm {dR}}(A/k)$. Equivalently, $\ker (\mathop {Fr}_A^{*}:H^{1}_{\mathrm {dR}}(A^{(1)}/\kappa )\to H^{1}_{\mathrm {dR}}(A/\kappa ))$ contains $\operatorname {{im}}(\mathop {Fr}_A^{*}:H^{1}_{\mathrm {dR}}(A/\kappa )=H^{1}_{\mathrm {dR}}(A^{(2)}/\kappa )\to H^{1}_{\mathrm {dR}}(A^{(1)}/\kappa ))=G_1^{(1)}$. The sum of the dimensions of these two vector space is $\dim _{\kappa } H^{1}_{\mathrm {dR}}(A/\kappa )=2g$ and the dimension of the image is equal to $g$, so the containment is in fact equality and we get $G_1=F^{1}$.

This gives a $G$-equivariant $\kappa$-linear isomorphism $H^{1}(A^{(1)},{\mathcal {O}})\simeq H^{0}(A,\Omega ^{1}_{A/\kappa })$. If $V$ is a representation of $G$ on a vector space over a field of characteristic $p$ then the character of Frobenius-twisted representation $V^{(1)}$ is equal to that of $V^{\tau }$ so the first assertion of the lemma follows.

(ii) By [Reference MessingMes72, Theorem V.1.10] the category of formal abelian schemes over $W(\kappa )$ is equivalent to the category of pairs $(A_0, \widetilde {F})$ where $A_0$ is an abelian variety over $\kappa$ and $\widetilde {F}$ is a $W(\kappa )$-submodule of $H^{1}_{\mathrm {cris}}(A_0/W(\kappa ))$ such that the quotient $H^{1}_{\mathrm {cris}}(A_0/W(\kappa ))/\widetilde {F}$ is torsion-free and $\widetilde {F}/p$ is equal to the first step of Hodge filtration $F^{1}\subset H^{1}_{\mathrm {dR}}(A/\kappa )=H^{1}_{\rm cris}(A/W(\kappa ))/p$.

Since the eigenvalues of $\sigma$ on $H^{1}_{\mathrm {cris}}(A/W(\kappa ))$ are pair-wise different, the same is true for the action of $\sigma$ on $H^{1}_{\mathrm {dR}}(A/\kappa )$ so there is a unique isomorphism $H^{1}_{\mathrm {dR}}(A/\kappa )\simeq H^{0}(A,\Omega ^{1}_{A/\kappa })\oplus H^{1}(A,{\mathcal {O}})$ of $G$-representations splitting the Hodge filtration on $H^{1}_{\mathrm {dR}}(A/\kappa )$. Since the isomorphism class of a representation of $G$ on a finite free $W(\kappa )$-module is completely determined by its reduction modulo $p$, there is a unique $G$-equivariant decomposition $H^{1}_{\mathrm {cris}}(A/W(\kappa ))=\widetilde {F}\oplus \widetilde {F}'$ lifting the decomposition of the de Rham cohomology. The formal lift ${\mathfrak A}$ of $A$ defined by $\widetilde {F}$ then comes equipped with a lift of the action of $G$ because $\widetilde {F}\subset H^{1}_{\mathrm {cris}}(A/W(\kappa ))$ is stable under $G$. The assertion about the action of $G$ on Hodge cohomology groups of ${\mathfrak A}$ is a formal consequence of (i).

Proof. Choose an ample line bundle $L$ on $B$. Then $M:={\bigotimes} _{h\in H}h^{*}(L)$ is an ample line bundle inducing a $H$-equivariant polarization $\lambda _{M}:B\to B^{\vee }$ that is separable because $\mathrm {char}\, F=0$. The induced map $\lambda _{M}^{*}:H^{0}(B^{\vee },\Omega ^{1}_{B^{\vee }/F})\to H^{0}(B,\Omega ^{1}_{B/F})$ provides a $H$-equivariant isomorphism between $H^{0}(B,\Omega ^{1}_{B/F})$ and $H^{1}(B,{\mathcal {O}})^{\vee }\simeq H^{0}(B^{\vee },\Omega ^{1}_{B^{\vee }/F})$

Proof. It is enough to prove the lemma for typical representations of rank $({l-1})/{2}$ because we have $H^{0}({\mathfrak B}_1\times _{{\mathcal {O}}_L}{\mathfrak B}_2,\Omega ^{1}_{{\mathfrak B}_1\times _{{\mathcal {O}}_L}{\mathfrak B}_2/{\mathcal {O}}_L})\simeq H^{0}({\mathfrak B}_1,\Omega ^{1}_{{\mathfrak B}_1/{\mathcal {O}}_L})\oplus H^{0}({\mathfrak B}_2,\Omega ^{1}_{{\mathfrak B}_2/{\mathcal {O}}_L})$ and likewise $H^{1}({\mathfrak B}_1\times _{{\mathcal {O}}_L}{\mathfrak B}_2,{\mathcal {O}})\simeq H^{1}({\mathfrak B}_1,{\mathcal {O}})\oplus H^{1}({\mathfrak B}_2,{\mathcal {O}})$ for all abelian schemes ${\mathfrak B}_1,{\mathfrak B}_2$, so the general case can be obtained by taking the products.

Fix a non-trivial root of unity $\zeta _l\in {\mathbb {Q}}(\mu _l)$ and an embedding ${\mathbb {Q}}(\mu _l)\subset W(k)[{1}/{p}]$, giving a bijection between roots of unity in ${\mathbb {Q}}(\mu _l)$ and $W(k)$. Each non-trivial root of unity $\zeta \in \mu _l(W(k))$ thus defines an embedding of the cyclotomic field $\varphi _{\zeta }:{\mathbb {Q}}(\mu _l)\to \bar {{\mathbb {Q}}}$ into the algebraic closure given by $\zeta _l\mapsto \zeta$.

Let $U$ be a typical representation of rank $({l-1})/{2}$ over $W(k)$. We associate to $U$ the set $\Phi =\{\varphi _{\zeta }|\chi _{\zeta ^{-1}}\text { appears in }U\}$ of embeddings. By the definition of typical representation $\Phi$ is a CM type of the field ${\mathbb {Q}}(\mu _l)$. Since ${\mathbb {Q}}(\mu _l)$ is Galois over ${\mathbb {Q}}$, the reflex field of $\Phi$ is contained in ${\mathbb {Q}}(\mu _l)$. By [Reference Chai, Conrad and OortCCO14, Corollary A.4.6.5] applied with $N=p$ (beware that the symbol ‘$p$’ bears its own meaning in this corollary) there exists an abelian variety $B'$ with complex multiplication by ${\mathbb {Q}}(\mu _l)$ of type $\Phi$ defined over an unramified extension $F$ of the reflex field that has good reduction at all primes above $p$. Take $L=F_{{\mathfrak p}}$ for some prime ${\mathfrak p}\subset {\mathcal {O}}_F$ above $p$ and consider the good model ${\mathfrak B}'$ of $B'$ over ${\mathcal {O}}_L$. The isogeny action of ${\mathbb {Q}}(\mu _l)$ then extends onto ${\mathfrak B}'$. In particular, embedding $G$ into ${\mathbb {Q}}(\mu _l)$ via $\sigma \mapsto \zeta _l$ gives an action of $G$ on ${\mathfrak B}'$ in the isogeny category. By our choice of $\Phi$ the $G$-representation $H^{0}({\mathfrak B}',\Omega ^{1}_{{\mathfrak B}'/{\mathcal {O}}_L})\otimes _{{\mathcal {O}}_L}L$ is isomorphic to $U\otimes _{W(k)}L$.

Analogously to the proof of Lemma 3.3, Serre's tensor product construction gives an isogeny ${\mathfrak B}'\to {\mathfrak B}$ to an abelian scheme ${\mathfrak B}$ that carries a genuine action of ${\mathbb {Z}}[\mu _l]$. In particular, $G$ acts on ${\mathfrak B}$ such that $H^{0}({\mathfrak B},\Omega ^{1}_{{\mathfrak B}/{\mathcal {O}}_L})$ is isomorphic to $U\otimes _{W(k)}{\mathcal {O}}_L$, as desired. To see that $H^{1}({\mathfrak B},{\mathcal {O}})$ is isomorphic to $U^{\vee }\otimes _{W(k)}{\mathcal {O}}_L$ it is enough to check the analogous statement for the generic fiber ${\mathfrak B}_L$ and this is given by Lemma 3.6.

Proof. Suppose on the contrary that these ranks of invariant subspaces are equal for all typical $U$. Using that $\Lambda ^{3}(V\oplus U)\simeq \Lambda ^{3} V\oplus \Lambda ^{2}V\otimes U\oplus V\otimes \Lambda ^{2} U\oplus \Lambda ^{3} U$ and the fact that twisting by an automorphism of $G$ does not change the invariant submodule of a representation we get

\[ \operatorname{{rk}} (\Lambda^{2}V\otimes U^{{\oplus} r})^{G} +\operatorname{{rk}} (V\otimes\Lambda^{2}(U^{{\oplus} r}))^{G} = \operatorname{{rk}}(\Lambda^{2} V^{\tau}\otimes U^{{\vee}{\oplus} r})^{G}+\operatorname{{rk}}(V^{\tau}\otimes\Lambda^{2} (U^{{\vee}{\oplus} r}))^{G} \]

for all typical $U$ and any multiplicity $r$ (by definition, the class of typical representations is closed under direct sums). Since $\Lambda ^{2}(U^{\oplus r})\simeq (\Lambda ^{2}U)^{\oplus r}\oplus (U^{\otimes 2})^{\oplus \binom {r}{2}}$ both sides of the equality are quadratic polynomials in $r$ for a fixed $U$, and comparing the leading coefficients we get

\[ \operatorname{{rk}}(V\otimes U^{{\otimes} 2})^{G}=\operatorname{{rk}}(V^{\tau}\otimes U^{{\vee}{\otimes} 2})^{G}. \]

As $V$ is assumed to be self-dual this equality is equivalent to $\operatorname {{rk}}(V\otimes U^{\otimes 2})^{G}=\operatorname {{rk}}(V^{\tau }\otimes U^{\otimes 2})^{G}$. In other words, the difference of characters $\chi _V-\chi _{V^{\tau }}$ is orthogonal to any character $\chi _{U^{\otimes 2}}$ for a typical $U$ with respect to the pairing $\langle \chi _1,\chi _2\rangle :=\sum _{g\in G}\chi _1(g)\chi _2(g)$ on the character ring $W(k)[G]$. Moreover, this implies that $\chi _V-\chi _{V^{\tau }}$ is orthogonal to $\chi _{U_1\otimes U_2}=\chi _{U_1}\cdot \chi _{U_2}$ for any two typical representations $U_1,U_2$ because $2\chi _{U_1\otimes U_2}=\chi _{(U_1\oplus U_2)^{\otimes 2}}-\chi _{U_1^{\otimes 2}}-\chi _{U_2^{\otimes 2}}$.

If $\zeta$ is any non-trivial root of unity, then for a set $\zeta _1,\ldots ,\zeta _{({l-3})/{2}}$ of pairwise different roots such that $\zeta _i\neq \zeta _j^{-1}$ and $\zeta _i\neq \zeta ^{\pm 1}$ for all $i,j$ both representations $\chi _{\zeta }\oplus \chi _{\zeta _1}\oplus \cdots \oplus \chi _{\zeta _{({l-3})/{2}}}$ and $\chi _{\zeta ^{-1}}\oplus \chi _{\zeta _1}\oplus \cdots \oplus \chi _{\zeta _{({l-3})/{2}}}$ are typical. Hence the span of characters of typical representations in $W(k)[G]$ contains all characters of the form $\chi _{\zeta }-\chi _{\zeta ^{-1}}$ for a non-trivial root of unity $\zeta$. By linearity of the pairing, $\chi _V-\chi _{V^{\tau }}$ is hence also orthogonal to $\chi ^{2}_{\zeta }+\chi ^{2}_{\zeta ^{-1}}=(\chi _{\zeta }-\chi _{\zeta ^{-1}})^{2}+2\chi _{1}$. However, $\langle \chi _V,\chi ^{2}_{\zeta }+\chi ^{2}_{\zeta ^{-1}}\rangle =2\langle \chi _V,\chi ^{2}_{\zeta }\rangle$ by self-duality of $V$ and likewise for $V^{\tau }$ so we get a contradiction since, by assumption, there exists $\zeta$ such that multiplicities of $\chi _{\zeta ^{-2}}$ in $V$ and $V^{\tau }$ are different.

Proof of Proposition 3.1 Let ${\mathfrak A}$ be a formal abelian scheme with an action of $G$ provided by Lemma 3.5(ii). Consider the representation $V=H^{0}({\mathfrak A},\Omega ^{1}_{{\mathfrak A}/W(\kappa )})$. We have $G$-equivariant isomorphisms $V\oplus V^{\tau }\simeq H^{0}({\mathfrak A},\Omega ^{1}_{{\mathfrak A}/W(\kappa )})\oplus H^{1}({\mathfrak A},{\mathcal {O}})\simeq H^{1}_{\mathrm {dR}}({\mathfrak A}/W(\kappa ))\simeq H^{1}_{\mathrm {cris}}(A/W(\kappa ))$, so every non-trivial character $\chi _{\zeta }$ appears in exactly one of $V$ and $V^{\tau }$ with multiplicity one. It follows that $V\simeq (V^{\tau })^{\tau }$. Since $V^{\vee }$ is obtained from $V$ by applying the twist $\tau$ an even number of times, the representation $V\otimes _{W(\kappa )}W(k)$ satisfies the assumptions of Lemma 3.10 and there exists a typical representation $U$ such that non-equality (5) holds. Let ${\mathfrak B}$ be an abelian scheme over ${\mathcal {O}}_L=W(k')$ provided by Lemma 3.8 applied to $U$. The product ${\mathfrak Z}'={\mathcal {A}}\times _{W(k)}{\mathfrak B}$ equipped with the diagonal action then has $H^{0}({\mathfrak Z}',\Omega ^{1}_{{\mathfrak Z}/{\mathcal {O}}_L})\simeq (V\oplus U)\otimes _{W(k)}{{\mathcal {O}}_L}$ and $H^{1}({\mathfrak Z}',{\mathcal {O}})\simeq (V^{\tau }\oplus U^{\vee })\otimes _{W(k)}{{\mathcal {O}}_L}$ as $G$-representations.

Since $H^{0}({\mathfrak Z}',\Omega ^{3}_{{\mathfrak Z}/{\mathcal {O}}_L})\simeq \Lambda ^{3} H^{0}({\mathfrak Z}',\Omega ^{1}_{{\mathfrak Z}'/{\mathcal {O}}_L})$ and $H^{3}({\mathfrak Z}',{\mathcal {O}})\simeq \Lambda ^{3} H^{1}({\mathfrak Z}',{\mathcal {O}})$ we get

\[ \operatorname{{rk}} H^{0}({\mathfrak Z}',\Omega^{3}_{{\mathfrak Z}/{\mathcal{O}}_L})^{G}\neq \operatorname{{rk}} H^{3}({\mathfrak Z}',{\mathcal{O}})^{G}. \]

If $\operatorname {{rk}} H^{0}({\mathfrak Z}',\Omega ^{3}_{{\mathfrak Z}/{\mathcal {O}}_L})^{G}> \operatorname {{rk}} H^{3}({\mathfrak Z}',{\mathcal {O}})^{G}$, then ${\mathfrak Z}={\mathfrak Z}'$ gives the desired inequality (4). If the inequality goes the other way, take ${\mathfrak Z}={\mathfrak Z}'^{\vee }$ using that there are $G$-equivariant isomorphisms $H^{0}({\mathfrak Z}'^{\vee },\Omega ^{1}_{{\mathcal {O}}_L})\simeq H^{1}({\mathfrak Z}',{\mathcal {O}})^{\vee }$ and $H^{1}({\mathfrak Z}'^{\vee },{\mathcal {O}})\simeq H^{0}({\mathfrak Z}',\Omega ^{1}_{{\mathfrak Z}'/{\mathcal {O}}_L})^{\vee }$.

Theorem 4.1 For every pair of distinct positive integers $i\neq j$ with $i+j\geq 3$ there exists a smooth proper formal scheme ${\mathfrak X}$ over ${\mathbb {Z}}_p$ with projective special fiber ${\mathfrak X}_{{\mathbb {F}}_p}$ such that the Hodge cohomology groups $H^{i}({\mathfrak X},\Omega ^{j}_{{\mathfrak X}/{\mathbb {Z}}_p})$ and $H^{j}({\mathfrak X},\Omega ^{i}_{{\mathfrak X}/{\mathbb {Z}}_p})$ are free ${\mathbb {Z}}_p$-modules of different rank.

Proof. Let $({\mathfrak U}_i)_{i\in I}$ be a finite cover of ${\mathfrak X}$ by affine open formal subschemes. Their generic fibers $U_i$ induce a covering of $X$ by affinoid subdomains. For any coherent sheaf ${\mathcal {F}}$ on ${\mathfrak X}$ its cohomology is computed by the Čech complex ${\bigoplus} _{i\in I}{\mathcal {F}}({\mathfrak U}_i)\to {\bigoplus} _{i,j\in I}{\mathcal {F}}({\mathfrak U}_i\cap {\mathfrak U}_j)\to \cdots$. The cohomology of the sheaf $F$ on $X$ associated to ${\mathcal {F}}$ is computed by the Čech complex ${\bigoplus} _{i\in I}F(U_i)\to {\bigoplus} _{i,j\in I}F(U_i\cap U_j)\to \cdots$ since higher cohomology of coherent sheaves on affinoids vanish. By definition, $F(U_i)={\mathcal {F}}({\mathfrak U}_i)[{1}/{p}]$ so the Čech complex on the generic fiber is obtained from that of the formal scheme by inverting $p$ and we get the isomorphism $H^{i}(X,F)\simeq H^{i}({\mathfrak X},{\mathcal {F}})$ which implies the statement by using ${\mathcal {F}}=\Omega ^{j}_{{\mathfrak X}/{\mathcal {O}}_K}$.

Proof. For each $n$, the quotient ${\mathfrak X}_n:={\mathfrak Y}_{{\mathcal {O}}_K/{\mathfrak m}^{n}}/\Gamma$ exists by [Reference GrothendieckSGA1, Expose 1, Proposition V.1.8] because admissibility of the action can be checked on the reduced subscheme which is projective in this case. The closed immersions ${\mathfrak Y}_{{\mathcal {O}}_K/{\mathfrak m}^{n}}\to {\mathfrak Y}_{{\mathcal {O}}_K/{\mathfrak m}^{n+1}}$ induce morphisms ${\mathfrak X}_n\to {\mathfrak X}_{n+1}$. Since the action is free, by [Reference GrothendieckSGA1, Corollarie V.2.4] the morphisms $\pi _n:{\mathfrak Y}_n\to {\mathfrak X}_n$ are étale with Galois group $\Gamma$ and the canonical maps ${\mathcal {O}}_{{\mathfrak X}_n}\otimes _{{\mathbb {Z}}}{\mathbb {Z}}[\Gamma ]\to \pi _{n*}{\mathcal {O}}_{Y_n}$ are isomorphisms.

Hence, the maps ${\mathfrak X}_n\to {\mathfrak X}_{n+1}$ are closed immersions inducing isomorphisms (this also follows from order of $\Gamma$ being invertible on ${\mathfrak Y}$, but is not true in general for a non-free action)

\[ {\mathfrak X}_{n+1}\times_{{\mathcal{O}}_{K}/{\mathfrak m}^{n+1}}{\mathcal{O}}_K/{\mathfrak m}^{n}\simeq {\mathfrak X}_n \]

and the schemes ${\mathfrak X}_n$ form the desired quotient formal scheme ${\mathfrak X}$. If $L$ is an ample line bundle on ${\mathfrak Y}_k$ then ${\bigotimes} _{\gamma \in \Gamma }\gamma ^{*}(L)$ descends to an ample line bundle on ${\mathfrak X}_k$ so ${\mathfrak X}_k$ is projective over $k$.

By the étaleness of the quotient map $\pi :{\mathfrak Y}\to {\mathfrak X}$ the canonical morphisms $\Omega ^{i}_{{\mathfrak X}/{\mathcal {O}}_K}\to (\pi _*\Omega ^{i}_{{\mathfrak Y}/{\mathcal {O}}_K})^{\Gamma }$ are isomorphisms and we get a Hochschild–Serre spectral sequence with the second page $E_2^{ab}=H^{a}(\Gamma , H^{b}({\mathfrak Y},\Omega ^{i}_{{\mathfrak Y}/{\mathcal {O}}_K}))$ converging to $H^{a+b}({\mathfrak X},\Omega ^{i}_{{\mathfrak X}/{\mathcal {O}}_K})$. Since higher cohomology of $\Gamma$ with coefficients in ${\mathcal {O}}_K$-modules vanishes, we get the desired isomorphism.

Proof of Theorem 4.1 We first treat the case of $i+j=3$, and the general case follows from the Lemma 4.3 below. Let ${\mathfrak Z}$ be a formal abelian scheme with an action of $G$ over the ring of integers ${\mathcal {O}}_L=W(k')$ in some unramified extension $L\supset {\mathbb {Q}}_p$ provided by Proposition 3.1. We would like to construct ${\mathfrak X}$ by taking the quotient of ${\mathfrak Z}$ by $G$, but first we need to make the action free in a way that does not spoil the discrepancy between ranks of invariant submodules. By [Reference RaynaudRay79, Proposition 4.2.3] or [Reference Bhatt, Morrow and ScholzeBMS18, § 2.2] there exists a smooth projective complete intersection ${\mathfrak Y}$ of dimension $\geq 4$ in projective space over ${\mathcal {O}}_L$ equipped with a free action of $G$. We equip the product ${\mathfrak Z}\times _{{\mathcal {O}}_L}{\mathfrak Y}$ with the diagonal action of $G$ which is free because the action on the second factor is free. Note that $H^{i}({\mathfrak Z}\times {\mathfrak Y},{\mathcal {O}})\simeq H^{i}({\mathfrak Z},{\mathcal {O}})$ and $H^{0}({\mathfrak Z}\times {\mathfrak Y},\Omega ^{i}_{{\mathfrak Z}\times {\mathfrak Y}/{\mathcal {O}}_L})\simeq H^{0}({\mathfrak Z},\Omega ^{i}_{{\mathfrak Z}/{\mathcal {O}}_L})$ for $i\leq 3$ by Künneth formula and Lefschetz hyperplane theorem [Reference Antieau, Bhatt and MathewABM19, Proposition 5.3] applied to ${\mathfrak Y}$.

First, we construct the scheme with desired property over the unramified extension ${\mathcal {O}}_L$ as the quotient ${\mathfrak X}':=({\mathfrak Z}\times {\mathfrak Y})/G$ provided by the Lemma 4.2. We have $H^{3}({\mathfrak X}',{\mathcal {O}})=H^{3}({\mathfrak Z}\times {\mathfrak Y},{\mathcal {O}})^{G}=H^{3}({\mathfrak Z},{\mathcal {O}})^{G}$ and $H^{0}({\mathfrak X}',\Omega ^{3}_{{\mathfrak X}'/{\mathcal {O}}_L})^{G}=H^{0}({\mathfrak Z},\Omega ^{3}_{{\mathfrak Z}/{\mathcal {O}}_L})^{G}$. Proposition 3.1 supplied ${\mathfrak Z}$ such that the ranks of invariant modules $H^{3}({\mathfrak Z},{\mathcal {O}})^{G}$ and $H^{0}({\mathfrak Z},\Omega ^{3}_{{\mathfrak Z}/{\mathcal {O}}_L})^{G}$ are different, so $H^{3}({\mathfrak X}',{\mathcal {O}})$ and $H^{0}({\mathfrak X}',\Omega _{{\mathfrak X}'/{\mathcal {O}}_L}^{3})$ are indeed of different rank. By Proposition 2.1 we have $3h^{3,0}({\mathfrak X}')+h^{2,1}({\mathfrak X}')=3h^{0,3}({\mathfrak X}')+h^{1,2}({\mathfrak X}')$ so the ranks of $H^{2}({\mathfrak X}',\Omega ^{1}_{{\mathfrak X}'/{\mathcal {O}}_L})$ and $H^{1}({\mathfrak X}',\Omega ^{2}_{{\mathfrak X}'/{\mathcal {O}}_L})$ are forced to be different as well.

Finally, the example over ${\mathbb {Z}}_p$ is obtained by Weil restriction of scalars ${\mathfrak X}:=\operatorname {{Res}}^{{\mathcal {O}}_L}_{{\mathbb {Z}}_p}{\mathfrak X}'$. Since ${\mathcal {O}}_L$ is finite étale over ${\mathbb {Z}}_p$ this is a smooth proper formal scheme over ${\mathbb {Z}}_p$ with projective special fiber ${\mathfrak X}_{{\mathbb {F}}_p}=\operatorname {{Res}}^{k'}_{{\mathbb {F}}_p}{\mathfrak X}_{k'}$. Since ${\mathfrak X}\times _{{\mathbb {Z}}_p}{\mathcal {O}}_L$ is isomorphic to $({\mathfrak X}')^{\times d}$ where $d$ is the degree of $k'$ over ${\mathbb {F}}_p$, the Hodge cohomology modules of ${\mathfrak X}$ are free and $h^{3,0}({\mathfrak X})-h^{0,3}({\mathfrak X})=d(h^{3,0}({\mathfrak X}')-h^{0,3}({\mathfrak X}'))$ because the symmetry on cohomology of ${\mathfrak X}'$ in degrees $1$ and $2$ holds by Corollary 2.2.

Lemma 4.3 Let ${\mathfrak X}$ be a smooth proper formal scheme over ${\mathcal {O}}_K$ such that $h^{3,0}({\mathfrak X})\neq h^{0,3}({\mathfrak X})$. Then for any $i,j\geq 0$ with $i\neq j$ and $i+j>3$ there exists a smooth projective scheme ${\mathfrak Y}$ over ${\mathcal {O}}_K$ such that $h^{i,j}({\mathfrak X}\times {\mathfrak Y})\neq h^{j,i}({\mathfrak X}\times {\mathfrak Y})$.

Proof Recall that for a formal scheme ${\mathfrak T}$ the number $\delta ^{i,j}({\mathfrak T})$ is defined as the difference $h^{i,j}({\mathfrak T})-h^{j,i}({\mathfrak T})$. Note that if ${\mathfrak Y}$ satisfies Hodge symmetry then

(6)\begin{equation} \delta^{i,j}({\mathfrak T}\times{\mathfrak Y})=\displaystyle\sum_{\substack{i_1+i_2=i\\ j_1+j_2=j}}\delta^{i_1,j_1}({\mathfrak T})h^{i_2,j_2}({\mathfrak Y}). \end{equation}

By Proposition 2.1 we have $\delta ^{2,1}({\mathfrak X})=-3\delta ^{3,0}({\mathfrak X})$ and $\delta ^{1,0}({\mathfrak X})=\delta ^{2,0}({\mathfrak X})=0$. Let us assume that $i>j$. For a scheme ${\mathfrak Y}$ denote by $H_{{\mathfrak Y}}(x,y):=\sum _{i,j}h^{i,j}({\mathfrak Y})x^{i}y^{j}$ its Hodge polynomial. If ${\mathfrak Y}_1\subset {\mathfrak Y}_2$ is a smooth closed subscheme of codimension $r+1$ in a smooth proper scheme, then the Hodge polynomial of the blow-up is given by

\[ H_{Bl_{{\mathfrak Y}_1}{\mathfrak Y}_2}(x,y)=H_{{\mathfrak Y}_2}(x,y)+H_{{\mathfrak Y}_1}(x,y)\cdot(xy+(xy)^{2}+\cdots+(xy)^{r}). \]

Let ${\mathfrak T}_{d,n}$ be a smooth degree $d$ hypersurface in ${\mathbb {P}}^{n+1}_{{\mathcal {O}}_K}$ where $d$ and $n$ are numbers that we will choose later. Construct a sequence of smooth projective schemes starting with ${\mathfrak Y}_0={\mathfrak T}_{d,n}$ and for $s\geq 0$ defining the scheme ${\mathfrak Y}_{s+1}$ as the blow-up of some projective space ${\mathbb {P}}^{N_s}_{{\mathcal {O}}_K}$ along some embedding ${\mathfrak Y}_s\subset {\mathbb {P}}^{N_s}_{{\mathcal {O}}_K}$ of codimension $\geq 2$ (the numbers $N_s>\dim _{{\mathcal {O}}_K}{\mathfrak Y}_s+1$ can be chosen arbitrarily). By the above formula we have $H_{{\mathfrak Y}_s}(x,y)=F(xy)+H_{{\mathfrak T}_{d,n}}(x,y)\cdot (xy)^{s}\cdot G(xy)$ where $F(t),G(t)\in 1+t{\mathbb {Z}}[t]$ are some polynomials.

Assume first that $i> j+3$. Then take $n=i-3-j,s=j$ and consider ${\mathfrak Y}:={\mathfrak Y}_s$ obtained by the above inductive procedure from ${\mathfrak T}_{d,n}$. We then have $h^{i-2,j-1}({\mathfrak Y})=h^{i-1,j-2}({\mathfrak Y})=h^{i,j-3}({\mathfrak Y})=0$ and $h^{i-3,j}({\mathfrak Y})=h^{n,0}({\mathfrak T}_{d,n})$. This turns (6) into

\[ \delta^{i,j}({\mathfrak X}\times{\mathfrak Y})=\delta^{3,0}({\mathfrak X})\cdot h^{i-3,j}({\mathfrak Y})+\sum_{i_2+j_2<i+j-3}\delta^{i-i_2,j-j_2}({\mathfrak X})\cdot h^{i_2,j_2}({\mathfrak Y}). \]

By [Reference Deligne and KatzSGA7, XI.Theoreme 1.5] the Hodge numbers $h^{a,b}({\mathfrak T}_{d,n})$ of a hypersurface are zero unless $a+b=n$ or $a=b$ and $h^{a,a}({\mathfrak T}_{d,n})=1$ for $0\leq a\leq n, 2a\neq n$. It follows that the Hodge polynomial $H_{{\mathfrak Y}}(x,y)$ is congruent modulo the ideal $(x,y)^{n+s}$ to a polynomial of the form $K(xy)$ with coefficients not depending on the degree $d$ of the hypersurface. In particular, $\delta ^{i,j}({\mathfrak X}\times {\mathfrak Y})$ is the sum of $\delta ^{3,0}({\mathfrak X})\cdot h^{i-3,j}({\mathfrak Y})=\delta ^{3,0}({\mathfrak X})\cdot h^{n,0}({\mathfrak T}_{d,n})$ and a number that does not depend on $d$. By [Reference Deligne and KatzSGA7, Corollaire 2.4] the number is $h^{n,0}({\mathfrak T}_{d,n})$ is equal to $\binom {d-1}{n+1}$ so for large enough $d$ this sum will be non-zero.

If $i=j+2$, then take ${\mathfrak Y}={\mathfrak Y}_s$ with $n=1,s=j-1$. An analogous computation gives that $\delta ^{i,j}({\mathfrak X}\times {\mathfrak Y})$ is the sum of a number that does not depend on $d$ and the expression

\[ \delta^{2,1}({\mathfrak X})\cdot h^{1,0}({\mathfrak T}_{d,1})+\delta^{3,0}({\mathfrak X})\cdot h^{0,1}({\mathfrak T}_{d,1})=\delta^{3,0}({\mathfrak X})(h^{0,1}({\mathfrak T}_{d,1})-3h^{1,0}({\mathfrak T}_{d,1})) ={-}2\delta^{3,0}({\mathfrak X})\binom{d-1}{2}, \]

which is non-zero for $d>2$.

Finally, if $i= j+3$ or $i=j+1$ we will be able to move asymmetry into higher cohomology just via Tate twists. Consider the scheme ${\mathfrak Y}=({\mathbb {P}}^{1}_{{\mathcal {O}}_K})^{d}$ where we will again choose $d$ at the end. We have

\[ \delta^{i,j}({\mathfrak X}\times{\mathfrak Y})=\sum_{r\leq ({i+j-3})/{2}}\delta^{i-r,j-r}({\mathfrak X})\cdot\binom{d}{r}. \]

This is a polynomial in $d$ with leading coefficient $\delta ^{i-r,j-r}({\mathfrak X})$ with $r=({i+j-3})/{2}$ which is nothing but $\pm \delta ^{3,0}({\mathfrak X})$ or $\pm \delta ^{2,1}({\mathfrak X})$ so this expression is non-zero for a large enough value of $d$.

Proof. Take $X$ to be the generic fiber of a formal scheme ${\mathfrak X}$ provided by Theorem 4.1. Applying Lemma 4.1 gives the result.

Proof. The proof of Theorem 4.1 provides us with a smooth proper scheme ${\mathfrak X}$ over ${\mathbb {Z}}_p$ with projective special fiber such that all Hodge groups $H^{i}({\mathfrak X},\Omega ^{j}_{{\mathfrak X}/{\mathbb {Z}}_p})$ are free over ${\mathbb {Z}}_p$ and $h^{3,0}({\mathfrak X})< h^{0,3}({\mathfrak X})$ while $h^{1,0}({\mathfrak X})=h^{0,1}({\mathfrak X})=0$ (the Hodge numbers in degree $1$ vanish because representations $U,V$ used in proof of Proposition 3.1 have trivial invariants). Since Hodge cohomology modules are free, the Hodge numbers of the special fiber ${\mathfrak X}_{{\mathbb {F}}_p}$ are equal to those of ${\mathfrak X}$.

We will now take the product of ${\mathfrak X}$ with an auxiliary projective scheme that will be constructed by approximating the stack $B(\mu _p\times {\mathbb {Z}}/p)$. By [Reference Antieau, Bhatt and MathewABM19, Theorem 1.2] applied over ${\mathbb {Z}}_p$ to the group scheme $\mu _p\times {\mathbb {Z}}/p$ with $d=3$ there exists a smooth projective scheme ${\mathfrak Y}'$ such that $H^{j}({\mathfrak Y}'_{{\mathbb {F}}_p},\Omega ^{i}_{{\mathfrak Y}'_{{\mathbb {F}}_p}})=H^{j}(B(\mu _p\times {\mathbb {Z}}/p)_{{\mathbb {F}}_p},\Omega ^{i})$ and $H^{j}({\mathfrak Y}',\Omega ^{i}_{{\mathfrak Y}'/{\mathbb {Z}}_p})=H^{j}(B(\mu _p\times {\mathbb {Z}}/p)_{{\mathbb {Z}}_p},\Omega ^{i})$ for pairs $i,j$ with $i+j\geq 3$ and $i=0$ or $j=0$. The Hodge polynomial $H_{B(\mu _p\times {\mathbb {Z}}/p)_{{\mathbb {F}}_p}}(x,y):=\sum _{i,j}h^{i,j}(B(\mu _p\times {\mathbb {Z}}/p)_{{\mathbb {F}}_p})x^{i}y^{j}$ of this stack is given by the product $H_{B\mu _{p,{\mathbb {F}}_p}}(x,y)H_{B({\mathbb {Z}}/p)_{{\mathbb {F}}_p}}(x,y)$. By [Reference TotaroTot18, Proposition 11.1] the first multiple $H_{B\mu _{p,{\mathbb {F}}_p}}$ is equal to $({1+x})/({1-xy})$ and $H_{B({\mathbb {Z}}/p)_{{\mathbb {F}}_p}}$ is equal to ${1}/({1-y})$ because the only non-zero Hodge cohomology groups of the classifying stack of a finite discrete group $\Gamma$ are given by group cohomology $H^{i}(B\Gamma _{{\mathbb {F}}_p},{\mathcal {O}})\simeq H^{i}(\Gamma ,{\mathbb {F}}_p)$. Take ${\mathfrak Y}:={\mathfrak Y}'\times {\mathfrak E}^{l}$ where ${\mathfrak E}$ is an elliptic curve over ${\mathbb {Z}}_p$ and $l$ is a number that we will choose at the end.

By the above computation, $H_{{\mathfrak Y}_{{\mathbb {F}}_p}}(x,y)$ is given by $({1}/({1-y}))\cdot (({1+x})/({1-xy}))\cdot (1+x+y+xy)^{l}$ modulo an element of the ideal $(x^{4},xy,y^{4})$. For the purpose of computing $h^{3,0}(({\mathfrak X}\times {\mathfrak Y})_{{\mathbb {F}}_p})$ and $h^{0,3}(({\mathfrak X}\times {\mathfrak Y})_{{\mathbb {F}}_p})$ we only care about this polynomial modulo $xy$ and we have $H_{({\mathfrak X}\times {\mathfrak Y})_{{\mathbb {F}}_p}}(x,y)\equiv (1+h^{2,0}({\mathfrak X}_{{\mathbb {F}}_p})x^{2}$ $+h^{3,0}({\mathfrak X}_{{\mathbb {F}}_p})x^{3}+h^{0,2}({\mathfrak X}_{{\mathbb {F}}_p})y^{2}+h^{0,3}({\mathfrak X}_{{\mathbb {F}}_p})y^{3})(1+(l+1)x+(\binom {l}{2}+l)x^{2}$ $+(\binom {l}{3}+\binom {l}{2})x^{3}+(l+1)y+(\binom {l}{2}+l+1)y^{2}+(\binom {l}{3}+\binom {l}{2}+l+1)y^{3})$ modulo the ideal $(x^{4},xy,y^{4})$. The Hodge numbers of the special fiber ${\mathfrak X}_{{\mathbb {F}}_p}\times {\mathfrak Y}_{{\mathbb {F}}_p}$ are thus given by $h^{3,0}(({\mathfrak X}\times {\mathfrak Y})_{{\mathbb {F}}_p})=\binom {l}{3}+\binom {l}{2}+h^{3,0}({\mathfrak X}_{{\mathbb {F}}_p})+(l+1)h^{2,0}({\mathfrak X}_{{\mathbb {F}}_p})$, $h^{0,3}(({\mathfrak X}\times {\mathfrak Y})_{{\mathbb {F}}_p})=\binom {l}{3}+\binom {l}{2}+l+1+h^{0,3}({\mathfrak X}_{{\mathbb {F}}_p})+(l+1)h^{0,2}({\mathfrak X}_{{\mathbb {F}}_p})$. Hence, $\delta ^{3,0}(({\mathfrak X}\times {\mathfrak Y})_{{\mathbb {F}}_p})=\delta ^{3,0}({\mathfrak X}_{{\mathbb {F}}_p})+l+1$. Since $\delta ^{3,0}({\mathfrak X}_{{\mathbb {F}}_p})<0$ there exists $l$ that makes this difference equal to zero.

The product ${\mathfrak X}\times {\mathfrak Y}$ is our desired scheme ${\mathfrak X}'$. Note that the Hodge numbers $h^{i,0}({\mathfrak Y}_{{\mathbb {Q}}_p})$ of the generic fiber ${\mathfrak Y}_{{\mathbb {Q}}_p}$ vanish for $i\leq 3$ so the asymmetry $h^{3,0}({\mathfrak X}')=h^{3,0}({\mathfrak X})\neq h^{0,3}({\mathfrak X})= h^{0,3}({\mathfrak X}')$ persists.

Proof. Let ${\mathfrak X}$ be a formal scheme provided by Theorem 4.1. Choose a point $x:\operatorname {{Spf}}{\mathbb {Z}}_p\to {\mathfrak X}$ (one sees easily that the proof of Theorem 4.1 can be modified to make sure that at least one point exists). Let $f:{\mathfrak X}'\to {\mathfrak X}$ be the blow-up of ${\mathfrak X}$ at the closed subscheme given by $x$. Note that $h^{i,j}({\mathfrak X}')=h^{i,j}({\mathfrak X})$ for $i\neq j$.

The special fiber ${\mathfrak X}'_{{\mathbb {F}}_p}$ is the blow-up of ${\mathfrak X}_{{\mathbb {F}}_p}$ at a point; denote by $E\subset {\mathfrak X}'_{{\mathbb {F}}_p}$ the exceptional divisor. Let $H$ be any ample divisor on ${\mathfrak X}'_{{\mathbb {F}}_p}$. We will find a linear combination $H+nE$ with $n\geq 0$ such that the top self-intersection $(H+nE)^{d}$ is prime to $p$. Since the sum of an ample divisor with an effective divisor is ample, this will provide an ample divisor with the desired property.

Suppose that $H|_E$ has degree $k$ in $\operatorname {{Pic}}(E)={\mathbb {Z}}$. Then $H^{d-i}\cdot E^{i}=k^{d-i}\cdot (-1)^{i}$ for $i>0$ by [Reference FultonFul98, Example 8.3.9.] and the projection formula, so

(7)\begin{equation} (H+nE)^{d}=H^{d}+\sum_{i=1}^{d}\binom{d}{i}n^{i}k^{d-i}({-}1)^{i-1}=H^{d}-(k-n)^{d}+k^{d}. \end{equation}

For varying $n$ the residue of $(k-n)^{d}$ modulo $p$ takes at least two different values, so we can find $n$ such that the expression (7) is prime to $p$.